Chebotarev Density Theorem
The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field 
that split in a certain way in an algebraic extension 
of 
. When the base field is the field 
of rational numbers, the theorem becomes much simpler.
Let 
be a monic irreducible polynomial of degree 
with integer coefficients with root 
, let 
, let 
be the normal closure of 
, and let 
be a partition 
of 
, i.e., an ordered set of positive integers 
with 
. A prime is said to be unramified (over the number field 
) if it does not divide the discriminant of 
. Let 
denote the set of unramified primes. Consider the set 
of unramified primes for which 
factors as 
modulo 
, where 
is irreducible modulo 
and has degree 
. Also define the density 
of primes in 
as follows:
![delta(S_P)=lim_(N->infty)(#<span style=]() {p in S_P:p<=N})/(#{p in S:p<=N}). " src="http://mathworld.wolfram.com/images/equations/ChebotarevDensityTheorem/NumberedEquation1.gif" style="height:38px; width:192px" /> |
Now consider the Galois group 
of the number field 
. Since this is a subgroup of the symmetric group 
, every element of 
can be represented as a permutation of 
letters, which in turn has a unique representation as a product of disjoint cycles. Now consider the set of elements 
of 
consisting of disjoint cycles of length 
, 
, ..., 
. Then 
.
As an example, let 
, so 
and 
, where 
is a primitive root of unity. Since 
has discriminant 
, the only ramified primes are 2 and 3.
Let 
be an unramified prime. Then 
has a root (mod 
) if and only if 2 has a cube root (mod 
), which occurs whenever 
(mod 3) or 
(mod 3) and 2 has multiplicative order modulo 
dividing 
. The first case occurs for half of all unramified primes and the second case occurs for one sixth of all primes. In the first case, 2 has a unique cube root modulo 
, so 
factors as the product of a linear and an irreducible quadratic factor mod 
. In the second case, 2 has three distinct cube roots mod 
, so 
has three linear factors mod 
. In the remaining case, which occurs for 1/3 of all unramified primes, 
is irreducible mod 
. Now consider the corresponding elements of 
. The first case corresponds to products of 2-cycles and 1-cycles (the identity), of which there are three, or half of the elements of 
, the second case corresponds to products of three 1-cycles, or the identity, of which there is just one element, or one sixth of the elements of 
, and the remaining case corresponds to 3-cycles, of which there are two, or one third the elements of 
. Since 
in this case, the Chebotarev density theorem holds for this example.
The Chebotarev density theorem can often be used to determine the Galois group of a given irreducible polynomial 
of degree 
. To do so, count the number of unramified primes up to a specified bound for which 
factors in a certain way and then compare the results with the fractions of elements of each of the transitive subgroups of 
with the same cyclic structure. Lenstra provides some good examples of this procedure.
REFERENCES:
Lenstra, H. "The Chebotarev Density Theorem." http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf.
